BERLIN, Sept 18 (Reuters) - German Chancellor Angela Merkel lauded her veteran finance minister Wolfgang Schaeuble at a birthday party in his honour on Monday, as a potential coalition partner laid claim to his ministry as their price for keeping her in power.

Speaking at Schaeuble’s 75th birthday celebration, Merkel paid tribute to his 45 years as a member of the German parliament, but gave few clues as to whether she wanted him to remain in his job after next Sunday’s election which she is on track to win.

“We’re fighting together at the moment to ensure there are at least four more years,” Merkel told attendees at the event hosted by her Christian Democrats (CDU) in Offenburg, in his constituency in the southwestern state of Baden-Wuerttemberg.

In a wheelchair since a deranged man shot him at an election campaign event a few days after German reunification in 1990, Schaeuble lives for his job and has made abundantly clear he would relish another run as finance minister.

But his fate may rest in the hands of the parties Merkel ends up governing with. On Monday, the pro-business Free Democrats (FDP), regarded as the more natural ally of her conservatives, made a claim to his ministry in exchange for joining the next coalition.

“The FDP should enter no government in which it cannot name a finance minister,” Alexander Hahn, a member of the FDP’s national executive, told mass-market daily Bild.

The FDP is still haunted by its decision not to take the finance ministry in Merkel’s second term - a move that backfired as Schaeuble quashed the FDP’s plans for tax cuts, infuriating its pro-business supporters who fled the party, booting it out of parliament in 2013.

The FDP has reinvented itself with a hard line on Europe and its leader has voiced concerns that Merkel and French President Emmanuel Macron plan to strengthen the euro zone by creating a system of financial transfers between states.

Instead, the FDP wants to do away with the euro zone’s bailout fund and make it easier for countries to leave the currency.

Schaeuble, who is the face of German austerity and has insisted that EU countries adhere to Europe’s budget rules, could help Merkel keep conservative voters and lawmakers on board as they press ahead with deeper euro zone integration.

Horst Seehofer, leader of Merkel’s Bavarian sister party, the Christian Social Union (CSU), told the Handelsblatt daily he favoured a tie-up with the FDP, who have also talked tough on migrants.

“I quite clearly hope for a black-yellow coalition,” he told the Handelsblatt daily referring to the parties’ colours.

Merkel, who was dressed in the conservatives’ black and the FDP’s yellow, paid tribute to Schaeuble as a European, a fighter for German unity, a passionate parliamentarian and an intellectual force who keeps calm in hectic situations.

But she also wished him time to spend with his family and to try out new things, as she presented him with the complete works of Dietrich Bonhoeffer, a German pastor and theologian known for his opposition to the Nazis.

“We wish you a good read, new insights, all the best and, in the name of the CDU and from me personally, dear Wolfgang Schaeuble many heartfelt thanks,” Merkel added.